Molecular Orbital Theory of Simple Diatomic Molecules and Ions
To infinity and beyong: diatomic molecules above the ... › assets › pdf › ams › 2014 ›...
Transcript of To infinity and beyong: diatomic molecules above the ... › assets › pdf › ams › 2014 ›...
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To ∞ and beyond: diatomic molecules above the dissociation limit"
David W. Schwenke"NASA Ames Research Center"
Applied Modeling & Simulation Seminar Series July 29, 2014
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Re-entry is Hot!!!"
To infinity and beyond 2
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Free
Stre
am
N2,
O2 T
=300
K
Pay
load
Car
bon
base
d H
eat S
hiel
d
Bou
ndar
y La
yer
N2,
O2,
NO
, CN
, C2
T~10
00K
Equ
ilibr
ium
Reg
ion
N, N
+ , O
, O+ ,
N2,
N2+ ,
NO
, e-
T~60
00K
Sho
ck
N, N
+ , O
, O+ ,
e-
T=20
000-
8000
0K
hν
v=~11 km/s
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HyperRad Development team"Alan Wray"Yen Liu"Duane Carbon"Winifred Huo"Galina Chaban"Richard Jaffe"David Schwenke"
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+
=
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Solar spectrum"
To infinity and beyond 7
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Spectrum for each compound is unique ! remote detection""Helium: solar in 1868, terrestrial in 1895"What causes the spectrum?"Newton didn’t know""
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Simplest spectrum: Hydrogen Atom"
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Visible Spectrum
Ultraviolet Spectrum 1880s:
1/λ=R[(1/n”)2-(1/n’)2]
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More complicated: Na"
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Quantum Mechanics"!! "
To infinity and beyond 11
ih ∂∂tΨα x , t( ) = − 1
2 ∇x2 +V x( )%& '(Ψα x , t( ) ≡ H x( )Ψα x , t( )
Ψα x , t( ) = e−iEαt/hψα x( )H x( )ψα x( ) = Eαψα x( )c = λνhνβα = Eβ − Eα
ψβ*∫ x( )M x( )ψα x( ) dx
2
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Quantum numbers"n: principle quantum number!!1,2,3,4,…!l: electronic angular momentum quantum number!!0 (S for sharp), 1 (P for principle), 2 (D for diffuse), 3 (F), 4 (G), …!Λ: electronic angular momentum along diatomic axis! 0 (Σ±), ±1 (Π), ±2 (Δ), ±3 (Φ), …!S: electron spin angular momentum quantum number!!0,1/2, 1, 3/2, …!Spin multiplicity: 2S+1!!1 (S=0, singlets), 2 (S=1/2, doublets), 3 (S=1, triplets), …!Interchange symmetry: g,u!!!!
To infinity and beyond 12
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Divide and Conquer"1: Freeze nuclei, solve for electron wave functions!! !high dimensionality, low density of states!2: Allow nuclei to move under forces of electron wave functions!! !low dimensionality, high density of states!Born and Oppenheimer 1927!H=Helec+Hnuc+Hcross!
To infinity and beyond 13
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Divide and Conquer"1: Freeze nuclei, solve for electron wave functions!! !high dimensionality, low density of states!2: Allow nuclei to move under forces of electron wave functions!! !low dimensionality, high density of states!Born and Oppenheimer 1927!H=Helec+Hnuc+Hcross!
!!
To infinity and beyond 14
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Divide and Conquer"1: Freeze nuclei, solve for electron wave functions!! !high dimensionality, low density of states!2: Allow nuclei to move under forces of electron wave functions!! !low dimensionality, high density of states!Born and Oppenheimer 1927!H=Helec+Hnuc+Hcross!
! !justify by low density of states, mN/me >>1!
To infinity and beyond 15
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Solving for electron motion"Integral-differential equation with homogenous boundary conditions at infinity!èexpand in terms of analytic basis functions!• Assign electrons to basis functions in all possible ways & diagonalize:!
!full configuration interaction (FCI)!!For 10 e-, 92 basis functions, !
• Or!• Optimize molecular orbitals with small FCI: 536 variables!
• Allow 0, 1, or 2 e- outside of small FCI space: 1,729,808 functions!
• Contract pair excitations: 47,375!
To infinity and beyond 16
925
!
"#
$
%&
2
≈ 2.5 ×1015
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-92.6
-92.55
-92.5
-92.45
-92.4
0 1 2 3 4 5 6
Tota
l Ene
rgy
(a.u
.)
Bond Length R (a.u.)
CN Potential Curves
1000/cm=10 micron
10,000/cm=1 micron
blue=4750 A
2Sig+2Pi
4Sig+
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-92.6
-92.5
-92.4
-92.3
-92.2
-92.1
0 1 2 3 4 5 6
Total
Energ
y (a.u
.)
Bond Length R (a.u.)
CN Potential Curves
1000/cm=10 micron
10,000/cm=1 micron
100,000/cm=1000 A
blue=4750 A
2Sig+2Pi
4Sig+4Del
4Pi4Sig-2Sig-2Del2Phi
6Sig+4Sig-
6Pi4Del4Phi
18
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Advances in potential energy curve calculation"A) Dynamic weighting for valence MO determination!
A) Adjust weighting by value of energy!
B) Rydberg Orbital Calculation!A) First compute valence MOs: inward!
B) Freeze valence MOs, compute Rydberg MOs: outward!
C) Diabatic States!A) Hcross remains small!
B) Phase factors!!
C) Splitting!
D) Easier to interpolate!
E) Easier to adjust!
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-150.2
-150.15
-150.1
-150.05
-150
-149.95
-149.9
-149.85
-149.8
1 2 3 4 5 6
Ener
gy in
Har
tree
Bond Length in Bohr
O2 1Sig+g
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Advances in potential energy curve calculation"A) Dynamic weighting for valence MO determination!
A) Adjust weighting by value of energy!
B) Rydberg Orbital Calculation!A) First compute valence MOs: inward!
B) Freeze valence MOs, compute Rydberg MOs: outward!
C) Diabatic States!A) Hcross remains small!
B) Phase factors!!
C) Splitting!
D) Easier to interpolate!
E) Easier to adjust!
To infinity and beyond 20
-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
1.5 2 2.5 3 3.5 4 4.5
Ener
gy in
Har
tree
Bond Length in Bohr
NO 2Pi
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Adiabatic vs. Diabatic states"Thermodynamics: without gain or loss of heat"Adiabatic from the Greek “incapable of being crossed” "" "Nuclei are stationary, solve for electron motion: "Diabatic – not adiabatic"""""" ""
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-129.8
-129.75
-129.7
-129.65
-129.6
-129.55
-129.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
Ener
gy (a
.u.)
Bond Length (a.u.)
NO 2Pi diabatic states
E1 0 H11 H12
= UT U
0 E2 H21 H22
H r( )ψαa = Eα r( )ψα
a
ψαd does not "change" with r
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Advances in potential energy curve calculation"A) Dynamic weighting for valence MO determination!
A) Adjust weighting by value of energy!
B) Rydberg Orbital Calculation!A) First compute valence MOs: inward!
B) Freeze valence MOs, compute Rydberg MOs: outward!
C) Diabatic States!A) Hcross remains small!
B) Phase factors!!
C) Easier to interpolate!
D) Easier to adjust!
E) Splitting!
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-129.8
-129.75
-129.7
-129.65
-129.6
-129.55
-129.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
Energ
y (a.u.)
Bond Length (a.u.)
NO 2Pi diabatic states
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-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
-129.2
1 2 3 4 5 6
Ener
gy in
Har
tree
Bond length in Bohr
NO 2Pi adiabatic/molecular diabatic states
To infinity and beyond 23
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-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
-129.2
-129.1
-129
1 2 3 4 5 6
Ener
gy in
Har
tree
Bond length in Bohr
Splitting NO 2Pi
To infinity and beyond 24
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-129.8
-129.7
-129.6
-129.5
-129.4
-129.3
-129.2
-129.1
-129
1 2 3 4 5 6
Ener
gy in
Har
tree
Bound length in Bohr
Split NO 2Pi
To infinity and beyond 25
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-75.4
-75.35
-75.3
-75.25
-75.2
-75.15
-75.1
-75.05
-75
1 2 3 4 5 6 7 8 9 10
Ene
rgy
(a.u
.)
Bond length (a.u.)
C2+
4sig-g2piu4pig
2delg2sig-g2sig+g4sig-u
2pig2sig+u4delu
4sig+u2sig-u2delu
6piu4piu
4delg4sig+g2phig
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-75.4
-75.35
-75.3
-75.25
-75.2
-75.15
-75.1
-75.05
-75
1 2 3 4 5 6 7 8 9 10
Ener
gy (a
.u.)
Bond length (a.u.)
C2+
3P
1D
1S
5S
4sig-g2piu4pig
2delg2sig-g2sig+g4sig-u
2pig2sig+u4delu
4sig+u2sig-u2delu
6piu4piu
4delg4sig+g2phig
To infinity and beyond 27
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Solving for nuclear motion"All states are coupled! (except ug)"Rotation strongly coupled to vibration"Finite Difference Boundary Value method""Bound states: diagonalize a banded matrix""Free states: solve banded linear equations""""
To infinity and beyond 28
0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7
Ener
gy (a
.u.)
Bond Length (a.u.)
N2 effective potential
200
100
50
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How to find energies"""""Bound state:"""Free state:"""
Hψα = Eαψα, ψα = ηη
∑ Fηα r( )
H = −h2
2µη
η
∑ ∂ 2
∂r2η + η
η !η
∑ Vη !η r( ) !η
limr→0Fηα r( ) = 0 and limr→∞Fη
α r( ) = 0b.c. determine energy
limr→0F "ηηE r( ) = 0
limr→∞F "ηηE r( ) = k "η
− 12 exp −ik "η r( )δη "η − exp ik "η r( ) Sη "η E( )%& '(
kη2 = 2µ E − eη( )
energy determines b.c.
incoming outgoing
complex
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How to find energies"""""Bound state:"""Free state:"""
Hψα = Eαψα, ψα = ηη
∑ Fηα r( )
H = −h2
2µη
η
∑ ∂ 2
∂r2η + η
η !η
∑ Vη !η r( ) !η
limr→0Fηα r( ) = 0 and limr→∞Fη
α r( ) = 0b.c. determine energy
limr→0F "ηηE r( ) = 0
limr→∞F "ηηE r( ) = k "η
− 12 exp −ik "η r( )δη "η − exp ik "η r( ) Sη "η E( )%& '(
kη2 = 2µ E − eη( )
energy determines b.c.
incoming outgoing
complex
acoustics: echoes, whisper rooms, …
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0
0.05
0.1
0.15
0.2
0.25
0.3
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Ener
gy in
Har
tree
Bond Lenth in Bohr
O2 Schumann-Runge
To infinity and beyond 31
3Σ-g
3Σ-u
1Πu
National Aeronautics and Space Administration! To infinity and beyond 32
1e-20
1e-15
1e-10
1e-05
1
100000
1e+10
48000 50000 52000 54000 56000 58000 60000
|S|^
2 in
arb
itrar
y un
its
Frequency omega in 1/cm
Γα / 2E − Eα( )2 + Γα
2 / 4
Odd Parity"
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Even Parity"
1e-20
1e-15
1e-10
1e-05
1
100000
1e+10
48000 50000 52000 54000 56000 58000 60000
|S|^
2 in
arb
itrar
y un
its
Frequency omega in 1/cm
To infinity and beyond 33
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0
0.05
0.1
0.15
0.2
0.25
0.3
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Ener
gy in
Har
tree
Bond Lenth in Bohr
O2 Schumann-Runge
To infinity and beyond 34
3Σ-g
3Σ-u
1Πu
National Aeronautics and Space Administration!
0
0.05
0.1
0.15
0.2
0.25
0.3
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Ener
gy in
Har
tree
Bond Lenth in Bohr
O2 Schumann-Runge
To infinity and beyond 35
3Σ-g
3Σ-u
1Πu
National Aeronautics and Space Administration!
-150.2
-150.15
-150.1
-150.05
-150
-149.95
-149.9
1 2 3 4 5 6 7
Ener
gy in
Har
tree
Bond Length in Bohr
03P+ 03P+
01D+ 01D+
01D+ 03P+
01S+ 03P+
1sig-u3delu
3sig+u3piu1piu5piu
3sig-u
To infinity and beyond 36
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Line shapes…"Bound-bound"" "Voigt profile is Gold-Standard"""Bound-free"
To infinity and beyond 37
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
γ / 2$ν − ν0( )2 + γ 2 / 4
d $ν
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
$γ / 2$ν − ν0( )2 + $γ 2 / 4
d $ν ,
$γ = γ + Γ
?
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Line shapes…"Bound-bound"" "Voigt profile is Gold-Standard"""Bound-free"
To infinity and beyond 38
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
γ / 2$ν − ν0( )2 + γ 2 / 4
d $ν
L ν( ) = A ν0( ) exp0
∞
∫ −ν − $ν( )2
2σ 2
%
&''
(
)**
$γ / 2$ν − ν0( )2 + $γ 2 / 4
d $ν ,
$γ = γ + Γ
? σ =!νc
kTm
A ν0( ) = 4ν03
3gc3hµ3
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Review"• Hypersonic Re-entry!
• Spectroscopy!
• Quantum Mechanics!
• Born-Oppenheimer Approximation!
• Electronic Structure Calculations!
• Diabatic States!
• Bound vs. Free states!
• Line shapes!
To infinity and beyond 39
-129.8
-129.75
-129.7
-129.65
-129.6
-129.55
-129.5
1.6 1.8 2 2.2 2.4 2.6 2.8 3
Ener
gy (a
.u.)
Bond Length (a.u.)
NO 2Pi diabatic states
1e-20
1e-15
1e-10
1e-05
1
100000
1e+10
48000 50000 52000 54000 56000 58000 60000
|S|^
2 in
arb
itrar
y un
its
Frequency omega in 1/cm